A grafting number^{[1]} is a number whose digits, represented in base b, appear before or directly after the decimal point of its p'th root. The simplest type of grafting numbers, where b=10 and p=2, deal with square roots in base 10 and are referred to as 2nd order base 10 grafting numbers.
Integers with this grafting property are called grafting integers (GIs).^{[2]} For example, 98 is a GI because:
The 2nd order base 10 GIs between 0 and 9999 are:
0 | 0 | 764 | 27.6405499... |
1 | 1 | 765 | 27.6586334... |
8 | 2.828427... | 5711 | 75.5711585... |
77 | 8.774964... | 5736 | 75.7363849... |
98 | 9.899495... | 9797 | 98.9797959... |
99 | 9.949874... | 9998 | 99.9899995... |
100 | 10.0 | 9999 | 99.9949999... |
More GIs that illustrate an important pattern, in addition to 8 and 764, are: 76394, 7639321, 763932023, and 76393202251. This sequence of digits corresponds to the digits in the following irrational number:
This family of GIs can be generated by Equation (1):
is called a grafting number (GN), and is special because every integer generated by (1) is a GI. For other GNs, only a subset of the integers generated by similar equations to (1) produce GIs.
Each GN is a solution for in the Grafting Equation (GE):
are integer parameters where is the grafting root, is the base in which the numbers are represented, is the amount the decimal point is shifted, and is the constant added to the front of the result.
When , all digits of represented in base will appear on both sides of the Equation (GE).
For the corresponding values are .
References
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